This section is mostly dedicated to the proof of Theorem REF .
We will start by shortly recalling the main idea of the proof of [1]}.
This will prepare the terrain for us here for the proof of the main results of the present paper.
The proof starts with key representation results for the Nash equilibrium and the mean field equilibrium.
In fact, let \(\hat{\alpha }^{i,N}\)  be a Nash equilibrium.
It follows from [1]} that there is a Lipschitz continuous function \(\widehat{\Lambda }:[0,T]\times \mathbb {R}^\ell \times \mathbb {R}^\ell \times \mathcal {P}_2(\mathbb {R}^\ell )\times \mathbb {R}\rightarrow \mathbb {A}\)  such that
\(\hat{\alpha }_t^{i,N} = \widehat{\Lambda }(t,X^{i,\underline{\hat{\alpha }}}_t, Y^{i,i}_t, L^N(\underline{X}_t^{\underline{\hat{\alpha }}}),\zeta ^N_t),\) 
